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Median-based estimation of the intensity of a spatial point process. (English) Zbl 1422.62299

Summary: This paper is concerned with a robust estimator of the intensity of a stationary spatial point process. The estimator corresponds to the median of a jittered sample of the number of points, computed from a tessellation of the observation domain. We show that this median-based estimator satisfies a Bahadur representation from which we deduce its consistency and asymptotic normality under mild assumptions on the spatial point process. Through a simulation study, we compare the new estimator, in particular, with the standard one counting the mean number of points per unit volume. The empirical study confirms the asymptotic properties established in the theoretical part and shows that the median-based estimator is more robust to outliers than standard procedures.

MSC:

62M30 Inference from spatial processes
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
62E20 Asymptotic distribution theory in statistics

Software:

spatstat
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References:

[1] Adell, J., Jodrá, P. (2005). The median of the Poisson distribution. Metrika, 61(3), 337-346. · Zbl 1079.62014
[2] Assunção, R., Guttorp, P. (1999). Robustness for inhomogeneous Poisson point processes. Annals of the Institute of Statistical Mathematics, 51, 657-678. · Zbl 0977.62032
[3] Baddeley, A., Turner, R. (2005). Spatstat: An R package for analyzing spatial point patterns. Journal of Statistical Software, 12, 1-42. · Zbl 0037.36701
[4] Baddeley, A., Turner, R., Møller, J., Hazelton, M. (2005). Residual analysis for spatial point processes (with discussion). Journal of the Royal Statistical Society: Series B (Statistical Methodology), 67(5), 617-666. · Zbl 1112.62302
[5] Berndt, S., Stoyan, D. (1997). Automatic determination of dendritic arm spacing in directionally solidified matters. International Journal of Materials Research (formerly Zeitschrift für Metallkunde), 88, 758-763.
[6] Byth, K. (1982). On robust distance-based intensity estimators. Biometrics, 38(1), 127-135. · doi:10.2307/2530295
[7] Clausel, M., Coeurjolly, J.-F., Lelong, J. (2015). Stein estimation of the intensity of a spatial homogeneous Poisson point process. Annals of Applied Probability (to appear). · Zbl 1345.60045
[8] Coeurjolly, J.-F., Lavancier, F. (2013). Residuals and goodness-of-fit tests for stationary marked Gibbs point processes. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 75(2), 247-276. · Zbl 07555447
[9] Coeurjolly, J.-F., Møller, J. (2014). Variational approach to estimate the intensity of spatial point processes. Bernoulli, 20(3), 1097-1125. · Zbl 1400.62208
[10] Daley, D. J., Vere-Jones, D. (2003). An introduction to the theory of point processes. Volume I: Elementary theory and methods (2nd ed.). New York: Springer. · Zbl 1026.60061
[11] David, H., Nagaraja, H. (2003). Order statistics (3rd ed.). Wiley, NJ: Hoboken. · Zbl 1053.62060
[12] Diggle, P. J. (2003). Statistical analysis of spatial point patterns (2nd ed.). London: Arnold. · Zbl 1021.62076
[13] Ghosh, J. (1971). A new proof of the Bahadur representation of quantiles and an application. Annals of Mathematical Statistics, 42(6), 1957-1961. · Zbl 0235.62006 · doi:10.1214/aoms/1177693063
[14] Guan, Y., Loh, J. M. (2007). A thinned block bootstrap procedure for modeling inhomogeneous spatial point patterns. Journal of the American Statistical Association, 102, 1377-1386. · Zbl 1332.62108
[15] Guan, Y., Sherman, M., Calvin, J. A. (2007). On asymptotic properties of the mark variogram estimator of a marked point process. Journal of Statistical Planning and Inference, 137(1), 148-161. · Zbl 1098.62121
[16] Guyon, X. (1991). Random fields on a network. New York: Springer. · JFM 11.0687.01
[17] Heinrich, L., Prokešová, M. (2010). On estimating the asymptotic variance of stationary point processes. Methodology and Computing in Applied Probability, 12(3), 451-471. · Zbl 1197.62122
[18] Ibragimov, I. A., Linnik, Y. V. (1971). Independent and stationary sequences of random variables. Groningen: Wolters-Noordhoff. · Zbl 0219.60027
[19] Illian, J., Penttinen, A., Stoyan, H., Stoyan, D. (2008). Statistical analysis and modelling of spatial point patterns. Statistics in practice. Chichester: Wiley. · Zbl 1197.62135
[20] Karáczony, Z. (2006). A central limit theorem for mixing random fields. Miskolc Mathematical Notes, 7, 147-160. · Zbl 1120.41301
[21] Lavancier, F., Møller, J., Rubak, E. (2014). Determinantal point process models and statistical inference. Journal of the Royal Statistical Society: Series B. doi:10.1111/rssb.12096. · Zbl 1414.62403
[22] Ma, Y., Genton, M., Parzen, E. (2011). Asymptotic properties of sample quantiles of discrete distributions. Annals of the Institute of Statistical Mathematics, 63(2), 227-243. · Zbl 1432.62035
[23] Machado, J., Santos Silva, J. (2005). Quantiles for counts. Journal of the American Statistical Association, 100(472), 1226-1237. · Zbl 1117.62395
[24] Magnussen, S. (2012). Fixed-count density estimation with virtual plots. Spatial Statistics, 2, 33-46. · doi:10.1016/j.spasta.2012.09.001
[25] Møller, J. (1994). Lectures on random Voronoi tessellations. New York: Springer. · Zbl 0812.60016 · doi:10.1007/978-1-4612-2652-9
[26] Møller, J., Waagepetersen, R. P. (2003). Statistical inference and simulation for spatial point processes. Boca Raton: Chapman and Hall/CRC. · Zbl 1039.62089
[27] Mrkvička, T., Molchanov, I. (2005). Optimisation of linear unbiased intensity estimators for point processes. Annals of the Institute of Statistical Mathematics, 57(1), 71-81. · Zbl 1083.62077
[28] Politis, D., Paparoditis, E., Romano, J. (1998). Large sample inference for irregularly spaced dependent observations based on subsampling. The Indian Journal of Statistics, Series A, 60(2), 274-292. · Zbl 1058.62549
[29] Prokešová, M., Jensen, E. (2013). Asymptotic Palm likelihood theory for stationary point processes. Annals of the Institute of Statistical Mathematics, 65(2), 387-412. · Zbl 1440.62343
[30] Redenbach, C., Särkkä, A., Sormani, M. (2015). Classification of points in superpositions of strauss and poisson processes. Spatial Statistics, 12, 81-95.
[31] Rose, C., Smith, M. (1996). The multivariate normal distribution. Mathematica Journal, 6(1), 32-37.
[32] Stevens, W. (1950). Fiducial limits of the parameter of a discontinuous distribution. Biometrika, 37(1-2), 117-129. · Zbl 0037.36701 · doi:10.1093/biomet/37.1-2.117
[33] Stoyan, D., Kendall, W. S., Mecke, J. (1995). Stochastic geometry and its applications (2nd ed.). Chichester: Wiley. · Zbl 0838.60002
[34] Van der Vaart, A. (2000). Asymptotic statistics (Vol. 3). Cambridge: Cambridge University Press. · Zbl 0910.62001
[35] Waagepetersen, R., Guan, Y. (2009). Two-step estimation for inhomogeneous spatial point processes. Journal of the Royal Statistical Society: Series B, 71, 685-702. · Zbl 1250.62047
[36] Zhengyan, L., Chuanrong, L. (1996). Limit theory for mixing dependent random variables (Vol. 378). Dordrecht: Kluwer Academic. · Zbl 0889.60001
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